Chapter 2: The Basic Concepts of Set Theory

1. Chapter 2: The Basic Concepts of Set Theory • 2.1 Symbols and Terminology • 2.2 Venn Diagrams and Subsets • 2.3 Set Operations and Cartesian Products • 2.4 Surveys and Cardinal Numbers

2. Section 2-2 • Venn Diagrams and Subsets

3. Venn Diagrams and Subsets • Venn Diagrams • Complement of a Set • Subsets of a Set • Proper Subsets • Counting Subsets

4. Venn Diagrams In set theory, the universe of discourse is called the universal set, typically designated with the letter U. Venn Diagrams were developed by the logician John Venn (1834 – 1923). In these diagrams, the universal set is represented by a rectangle and other sets of interest within the universal set are depicted as circular regions.

5. A U Venn Diagrams The rectangle represents the universal set, U, while the portion bounded by the circle represents set A.

6. A U Complement of a Set The colored region inside U and outside the circle is labeled A'(read “Aprime”). This set, called the complement of A, contains all elements that are contained in U but not in A.

7. Complement of a Set For any set A within the universal set U, the complement of A, written A',is the set of all elements of U that are not elements of A. That is

8. B A U Subsets of a Set Set A is a subset of set B if every element of A is also an element of B. In symbols this is written

9. Example: Subsets Fill in the blank with to make a true statement. a) {a, b, c} ___ { a, c, d} b) {1, 2, 3, 4} ___ {1, 2, 3, 4} Solution a) {a, b, c} ___ { a, c, d} b) {1, 2, 3, 4} ___ {1, 2, 3, 4}

10. Set Equality (Alternative Definition) Suppose that A and B are sets. Then A = B if and only if

11. Proper Subset of a Set Set A is a proper subset of set B if In symbols, this is written

12. Example: Proper Subsets Decide whether or both could be placed in each blank to make a true statement. a) {a, b, c} ___ { a ,b, c, d} b) {1, 2, 3, 4} ___ {1, 2, 3, 4} Solution a) both b)

13. Counting Subsets One method of counting subsets involves using a tree diagram. The figure below shows the use of a tree diagram to find the subsets of {a, b}. Yes No {a, b} {a} {b} Yes No Yes No

14. Number of Subsets The number of subsets of a set with n elements is 2n. The number of proper subsets of a set with n elements is 2n – 1.

15. Example: Number of Subsets Find the number of subsets and the number of proper subsets of the set {m, a, t, h, y}. Solution Since there are 5 elements, the number of subsets is 25 = 32. The number of proper subsets is 32 – 1 = 31.